Stages of Math Development: How to Plan for Intervention


Hey friends! Today we are talking about the different stages of math development and how understanding these three stages can help you better plan for math interventions!! Fun stuff, right?

The Three Stages

Do you ever feel frustrated that you have thoroughly taught a concept but some of your students just aren't getting it? Two weeks of teaching place value and your students are still counting all of the cubes in a stick of ten can definitely make you feel like a teacher failure...trust me, I've been there.

This past year I was introduced to the concept of the stages of math development, which are:



Stage 1: Concrete

The concrete stage is all about getting your students hands on some great math manipulatives. We sometimes forget that some of our students have never been exposed to these math concepts. Letting them explore and put their hands on a manipulative can go A LONG WAY to helping them with a math concept. 




Place Value Example: 
-Start by giving students cubes and building numbers to 20. You will be surprised by long it will take some students to understand the concept of grouping cubes into "sticks of ten". 

-Once students have mastered place value to 20 with cubes, I introduce base ten blocks. We practice building numbers EVERY DAY during our math warm ups, then I continue to build with my struggling students during their guided math small group. 

Stage 2: Representational

The representational stage is where I start having my students draw objects to represent the math concept. This stage is much simpler because it requires less materials and can be done in so many different ways. This stages starts to transition your students into the independent stage of the math concept. 





Place Value Example: 
-To transition to representational, I start by having students build a number with their base ten blocks. Then, I tell students to draw what they see. I slowly start taking away the manipulatives until they can successfully draw the math concept on their own. 


Stage 3: Abstract

The abstract stage is typically and independent stage of math development. Students are able to use the math concept with little teacher assistance because it mostly consists of mental math. I may teach concepts from this stage once or twice at my small group table to check for understanding, but I mostly have students practice this stage in stations and other activities. 


Place Value Example:
-Once students have mastered drawing base ten place value, the next step is to be able to construct/deconstruct expanded form without seeing the base ten blocks. This is a fundamental skill that will help them when adding larger numbers using open number lines in the future. 

Planning for Math Intervention

How does all of this tie into he daunting task of math intervention? Easy, students do not move up in stages until they have fully mastered each stage. 

For example, if a student cannot successfully show the number 45 with base ten blocks, then they do not have a full understanding of number sense and SHOULD NOT move on to the representational stage of drawing the number 45 using base ten pictures.

Another example, if a student cannot successfully draw the base ten pictures and count the number 45 as 10, 20, 30, 40, 41, 42, 43, 44, 45, then they are not ready to move on to expanded form yet. 

Teacher's Secret: Stop trying to move so fast! I know what you're thinking, "But it's my job to teach ALL of my standards!! I already don't have enough time, and you're wanting me to slow down??" Here's the thing, I still teach ALL of my standards in my whole group mini-lessons. Are all of my students successful in those lessons? No. BUT, in my small group math time is where I slow down and meet my students where they are. Do I want Little Johnny to master expanded form by second grade? Yes!! Can he do that if he never fully understood the value of a number? No! So I slow down and meet him where he is and give him the best foundation I can and his best chance at being successful in the future. 

Classroom Tested "Concrete" Math Strategies

Here are a few kid-tested and approved concrete strategies that we use in our classroom: 








This activity explained here: 


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Easy Prep Math Stations

Need some easy prep, low explanation math stations? Check these out!






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